Single star-based orientation method using dual-axis level sensor

ABSTRACT

Disclosed is a single star-based orientation method using a dual-axis level sensor, which includes a calibration process and an actual calculation process.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority from Chinese Patent Application No. 202111348637.6, filed on Nov. 15, 2021. The content of the aforementioned applications, including any intervening amendments thereto, is incorporated herein by reference in its entirety.

TECHNICAL FIELD

This application relates to absolute attitude measurement, and more particularly to a single star-based orientation method using a dual-axis level sensor.

BACKGROUND

Servo star sensors are designed to enable the all-day observation during the measurement of astronomical position and attitude on the ground. By means of the small view field of the servo star sensors, the celestial objects are observable in the daytime, and the observation of astronomical objects is changed from passive capture to active star-seeking. The active star-seeking process includes astronomical prediction, motion compensation and image stabilization-based tracking, and is difficult to implement. According to the principle of attitude solution, it is necessary to observe at least two non-parallel space vectors to obtain the attitude information. In view of this, with regard to the servo star sensors, the observation target needs to be continuously changed to achieve the attitude measurement.

Regarding the current servo star sensor-based orientation, there is still a lack of a single star observation-based orientation strategy, and the dual-axis level sensor information-based fusion method is also absent.

For the existing servo star sensors, the active star-seeking is usually adopted for the attitude measurement, in which a priori information is introduced to predict the orientation of 3 to 5 celestial objects, and a servo is guided by the outline attitude information for observation. The active star-seeking has acceptable reliability under quasi-static conditions; under dynamic conditions, affected by the carrier motion, the later observation targets are prone to fall out of the observation field of view. Though some star sensors use inertial components such as gyroscopes to compensate for attitude changes of the carrier during the observation, the small observation view field still fails to enable the reliable observation.

A miss distance is used in a control strategy of the single star tracking for control feedback, which greatly enhances the control robustness. This method is often used in some geodetic astronomical surveys for orientation, but the fine adjustment of the measuring instrument is strictly required, which is difficult to achieve for dynamic carriers.

In summary, there is still a lack of a convenient single star-based orientation method that can reach the stable and reliable orientation with high update rate under dynamic conditions. A multi-star simultaneous solution is not only complicated in control strategy, but also difficult in the conversion of the calculation reference system, failing to obtain high-precision orientation information.

SUMMARY

In order to solve the above-mentioned problems, the present disclosure provides a single star-based orientation method using a dual-axis level sensor, which enables the orientation survey with a single astronomical object and improves a data update rate and measurement accuracy of astronomical orientation by measuring a redundant non-level information of the dual-axis level sensor.

Technical solutions of this application are described as follows.

This application provides a single star-based orientation method using a dual-axis level sensor, comprising:

(S1) denoting any two orthogonal side surfaces of a hexahedron on a star sensor as a first side surface and a second side surface, respectively; denoting a component of a vector ν_(right) on an X_(n)-axis of a quasi-horizontal reference system as t₄; and denoting a component of the vector ν_(right) on a Z_(n)-axis of the quasi-horizontal reference system as t₆;

wherein the vector ν_(right) is obtained by rotating a normal vector of the second side surface around the Z_(n)-axis of the quasi-horizontal reference system by a preset azimuth angle θ_(dir) ^(right);

(S2) calculating the t₄ and t₆ through the following equations:

$\left\{ {\begin{matrix} {t_{4} = {- \left( {{\sin\;{{\alpha sin}\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)}} + {\cos\;{{\alpha sin}\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)}}} \right)}} \\ {t_{6} = \left( {{\cos\;{{\alpha sin}\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)}} - \ {\sin\;{{\alpha sin}\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)}}} \right)} \end{matrix};} \right.$

wherein α is an angle between an X_(s)-axis of a dual-axis level sensor reference system and an X-axis of a hexahedron reference system on the star sensor; θ_(x) ^(pitch) is a measurement reading of the dual-axis level sensor on the X_(s)-axis; O_(y) ^(pitch) is a measurement reading of the dual-axis level sensor on a Y_(s)-axis of the dual-axis level sensor reference system; Δθ_(x) ^(pitch) is a bias of the dual-axis level sensor on the X_(s)-axis; and Δθ_(y) ^(pitch) is a bias of the dual-axis level sensor on the Y_(s)-axis;

(S3) denoting an observation vector of a single astronomical object in the hexahedron reference system as ν_(PRI) and denoting a reference vector of the single astronomical object in an inertial reference system as ν_(GND), wherein the reference vector of the single astronomical object is obtained from a star catalog, and the ν_(GND) and ν_(PRI) satisfy the following equation:

R _(z)(θ_(z))·R _(x)(θ_(x))·R _(y)(θ_(y))·ν_(GND)=ν_(PRI);

wherein R_(y)(θ_(y)) denotes that the reference vector ν_(GND) rotates around a Y-axis of the hexahedron reference system by an azimuth angle θ_(y); R_(x)(θ_(x)) denotes that the reference vector ν_(GND) rotates around the X-axis of the hexahedron reference system by a pitch angle θ_(x); and R_(z)(θ₂) denotes that the reference vector ν_(GND) rotates around a Z-axis of the hexahedron reference system by a roll angle θ_(z);

(S4) calculating the pitch angle θ_(x) and the roll angle θ_(z) according to the t₄ and t₆ through the following equations:

$\left\{ {\begin{matrix} {\theta_{x} = {{- \arcsin}t_{6}}} \\ {\theta_{z} = {- {\arcsin\left( {{t_{4}/\cos}\theta_{x}} \right)}}} \end{matrix};} \right.$

and

(S5) letting ν_(GND0)=R_(x)(−θ_(x))·R₂(−θ₂)·θ_(PRI) and obtaining ν_(GND)=R_(y)(−θ_(y))·ν_(GND0); and calculating the azimuth angle θ_(y) through the following equation to complete a single star-based orientation:

${\theta_{y} = {\arctan 2\left( {\frac{{v\;{1 \cdot v}\; 6} - {v\;{3 \cdot v}\; 4}}{{v\; 4^{2}} + {v\; 6^{2}}},\ \frac{{v\;{1 \cdot v}\; 4} + {v\;{3 \cdot v}\; 6}}{{v\; 4^{2}} + {v\; 6^{2}}}} \right)}};$

wherein [ν1 ν2 ν3]=ν_(GND); ν1-ν3 are components of the ν_(GND) on three axes of the inertial reference system, respectively; [ν4 ν5 ν6]^(T)=ν_(GND0); and ν4-ν6 are components of the ν_(GND0) on the X_(n)-axis, the Y_(n)-axis and the Z_(n)-axis, respectively.

In some embodiments, in the step (S1), the preset azimuth angle θ_(dir) ^(right) is set through steps of:

defining the X_(n)-axis of the quasi-horizontal reference system as a projection of the X-axis of the hexahedron reference system on a horizontal plane; defining a Y_(n)-axis of the quasi-horizontal reference system such that the Y_(n)-axis is orthogonal to the X_(n)-axis on the horizontal plane and a Z_(n)-axis of the quasi-horizontal reference system satisfies a right-hand rule, and has a general upward pointing direction;

measuring, by a first theodolite and a second theodolite, a normal pitch angle θ_(pitch) ^(front) of the first side surface and a normal pitch angle θ_(pitch) ^(right) of the second side surface, respectively; wherein in the quasi-horizontal reference system, a normal vector ν_(front) of the first side surface is expressed as:

${v_{front} = {{{R_{y}\left( \theta_{pitch}^{front} \right)} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} {{vf}\; 1} \\ 0 \\ {{vf}\; 3} \end{bmatrix}}};$

wherein R_(y)(θ_(pitch) ^(front)) denotes that a vector [0 0 1] rotates around a Y_(n)-axis of the quasi-horizontal reference system by the normal pitch angle θ_(pitch) ^(front); and νƒ1,0 and νƒ3 are components of the normal vector ν_(front) on the X_(n)-axis, Y_(n)-axis and Z_(n)-axis, respectively;

a normal vector ν _(right) of the second side surface without rotating by the preset azimuth angle θ_(dir) ^(right) is expressed as:

${{\overset{\_}{v}}_{right} = {{{R_{y}\left( \theta_{pitch}^{right} \right)} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} {{vf}\; 1} \\ 0 \\ {{vf}\; 3} \end{bmatrix}}};$

wherein R_(y)(θ_(pitch) ^(right)) denotes that the vector [00 1] rotates around the Y_(n)-axis of the quasi-horizontal reference system by the normal pitch angle θ_(pitch) ^(right); and νr1,0 and νr3 are components of the normal vector ν _(right) on the X_(n)-axis, Y_(n)-axis and Z_(n)-axis, respectively;

-   -   supposing that the vector ν_(right) is obtained by rotating the         vector ν _(right) by the preset azimuth angle θ_(dir) ^(right)         and expressing the vector ν_(right) as follows:

${v_{right} = {{{R_{z}\left( \theta_{dir}^{right} \right)} \cdot \begin{bmatrix} {{vr}\; 1} \\ 0 \\ {{vr}\; 3} \end{bmatrix}} = \begin{bmatrix} {{vr}\;{1 \cdot \cos}\;\theta_{dir}^{right}} \\ {{- {vr}}\;{1 \cdot \sin}\;\theta_{dir}^{right}} \\ {{vr}\; 3} \end{bmatrix}}};$

wherein R_(z)(θ_(dir) ^(right)) denotes that a vector [νr 0 νr3] rotates around the Z_(n)-axis by the preset azimuth angle θ_(dir) ^(right); and

according to an orthogonal relationship between the normal vector ν_(front) and the normal vector ν_(right), obtaining the following equation:

$\left( {\theta_{dir}^{right} = {a{\cos\left( \frac{{- v}f\;{3 \cdot {vr}}\; 3}{{vf}\;{1 \cdot {vr}}\; 1} \right)}}} \right)^{.}$

In some embodiments, an axis of a measuring lens barrel of the first theodolite coincides with a normal of the first side surface of the hexahedron, and an axis of a measuring lens barrel of the second theodolite coincides with a normal of the second side surface of the hexahedron.

In some embodiments, in the step (S2), the angle α, the bias Δθ_(x) ^(pitch) and the bias Δθ_(y) ^(pitch) are obtained through steps of:

defining the X_(n)-axis of the quasi-horizontal reference system as a projection of the X-axis of the hexahedron reference system on a horizontal plane; defining a Y_(n)-axis of the quasi-horizontal reference system such that the Y_(n)-axis is orthogonal to the X_(n)-axis on the horizontal plane and the Z_(n)-axis of the quasi-horizontal reference system satisfies a vertically-upward right-hand rule; allowing the dual-axis level sensor reference system to be coplanar with the hexahedron reference system; and expressing a direction vector of the X_(s)-axis in the hexahedron reference system as [cos α 0−sin α] and a direction vector of the Y_(s)-axis in the hexahedron reference system as [−sin α 0−cos α];

denoting a normal vector of the first side surface as ν_(front); and obtaining a normal vector ν_(up) of a third side surface orthogonal to both the first side surface and the second side surface by cross product, expressed as:

ν_(up)=ν_(right)×ν_(front);

obtaining a transformation matrix R_(GND2PRI) from the quasi-horizontal reference system to the hexahedron reference system, expressed as:

R _(GND2PRI)=[ν_(front)ν_(right)ν_(up)];

obtaining a transpose matrix of the transformation matrix R_(GND2PRI), expressed as:

${R_{{PRI}2{GND}} = {R_{{GND}2{PRI}}^{T} = \begin{bmatrix} t_{1} & t_{2} & t_{3} \\ t_{4} & t_{5} & t_{6} \\ t_{7} & t_{8} & t_{9} \end{bmatrix}}};$

wherein T represents a transposition operation; and t₁-t₉ are elements of the transformation matrix R_(GND2PRI);

acquiring a direction vector ν_(x), of the X_(S)-axis in the quasi-horizontal reference system, expressed as:

${v_{X_{s}} = \begin{bmatrix} {{t_{1}\cos\alpha} - {t_{3}\sin\alpha}} \\ {{t_{4}\cos\alpha} - {t_{6}\sin\alpha}} \\ {{t_{7}\cos\alpha} - {t_{9}\sin\alpha}} \end{bmatrix}};$

acquiring a direction vector ν_(y), of the Y_(s)-axis in the quasi-horizontal reference system, expressed as:

${v_{Y_{s}} = \begin{bmatrix} {{{- t_{1}}\sin\alpha} - {t_{3}\cos\alpha}} \\ {{{- t_{4}}\sin\alpha} - {t_{6}\cos\alpha}} \\ {{{- t_{7}}\sin\alpha} - {t_{9}\cos\alpha}} \end{bmatrix}};$

building relations between t₄ and t₆ according to the direction vector ν_(x), and the direction vector ν_(y), expressed as:

$\left\{ {\begin{matrix} {{\sin\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)} = {{t_{4}\cos\alpha} - {t_{6}\sin\alpha}}} \\ {{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)} = {{{- t_{4}}\sin\alpha} - {t_{6}\cos\alpha}}} \end{matrix};} \right.$

according to the relations between t₄ and t₆, obtaining a relation between the measurement reading θ_(x) ^(pitch) and the bias Δθ_(x) ^(pitch) and a relation between the measurement reading θ_(y) ^(pitch) and the bias Δθ_(y) ^(pitch), expressed as:

$\left\{ {\begin{matrix} {\theta_{x}^{pitch} = {{\Delta\theta_{x}^{pitch}} - {\arcsin\left( {{t_{4}\cos\alpha} - {t_{6}\sin\alpha}} \right)}}} \\ {\theta_{y}^{pitch} = {{\Delta\theta_{y}^{pitch}} - {\arcsin\left( {{{- t_{4}}\sin\alpha} - {t_{6}\cos\alpha}} \right)}}} \end{matrix};} \right.$

changing a two-axis level degree of the dual-axis level sensor to obtain (θ_(x) ^(pitch), θ_(y) ^(pitch)) and (t₄, t₆) under different two-axis level degrees; and plugging the (θ_(x) ^(pitch), θ_(y) ^(pitch)) and (t₄, t₆) under different two-axis level degrees into the relation between the measurement reading θ_(x) ^(pitch) and the bias Δν_(x) ^(pitch) and the relation between the measurement reading, θ_(y) ^(pitch) and the bias Δθ_(y) ^(pitch) to obtain the angle α, the bias Δθ_(x) ^(pitch) and the bias Δθ_(y) ^(pitch);

In some embodiments, the first side surface is a front surface of the hexahedron, and the second side surface is a right surface of the hexahedron.

Compared to the prior art, this application has the following beneficial effects.

1. By means of the two-axis level degree measured by the dual-axis level sensor, the method provided herein enables a continuous orientation survey of a single astronomical object and overcomes problems caused by astronomical predictions and attitude changes when the servo star sensor switches between multiple astronomical objects, so as to allow a long-term and continuous orientation survey with high update rate based on miss distance after successfully capturing one astronomical object, eliminating the need for simultaneous tracking of multiple astronomical objects and reducing the number of observed astronomical objects. The orientation method provided herein is particularly suitable for the observation and tracking of a single target in the day time, and can also use obvious targets such as the sun for attitude measurement, simplifying the photoelectric detection and enhancing the data update rate and measurement accuracy for the astronomical orientation.

2. The orientation method provided herein proposes a simple and effective mathematical modeling strategy of the reference system, which greatly reduces the complexity of calibration parameters. Specifically, when constructing the quasi-horizontal reference system, one axis is allowed to coincide with a projection of a normal vector of one side surface of the hexahedron reference system to omit an orientation variable. When constructing a relationship between the dual-axis level sensor reference system and the hexahedron reference system, the relationship is simplified to an orientation error, and the two-axis non-levelness error is incorporated into the bias of a level reading to reduce the data collection required for calibration.

3. In the orientation method provided herein, the hexahedron reference system is measured by means of the autocollimation of the theodolite, and the quasi-horizontal reference system is measured by the dual-axis level sensor. Then structural parameters of the quasi-horizontal reference system and the hexahedron reference system can be calibrated according to the pitch angle measured by the theodolite and the reading of the dual-axis level sensor. Accordingly, with the help of the theodolite, the collection of all calibrated data can be completed, simplifying the calibration process.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a single star-based orientation method using a dual-axis level sensor;

FIG. 2 schematically depicts a relationship between a dual-axis level sensor reference system, a hexahedron reference system and a quasi-horizontal reference system; and

FIG. 3 schematically depicts a calibration of the dual-axis level sensor reference system and the quasi-horizontal reference system.

In the drawings: 1, servo star sensor; 2, hexahedron reference; 3, first theodolite; 4, second theodolite; and 5, data recording computer.

DETAILED DESCRIPTION OF EMBODIMENTS

The disclosure will be clearly and completely described below with reference to the accompanying drawings and embodiments.

As shown in the flow chart in FIG. 1, a single star-based orientation method using a dual-axis level sensor includes a first calibrating process, a second calibrating process and an actual calculating process. A key of the single star-based orientation is to project an observed single astronomical object onto a “virtual” horizontal plane (quasi-horizontal plane) via a dual-axis level sensor to obtain an azimuth angle of the projection, which is then compared with that in an inertial reference system. An azimuth angle difference is a current measured azimuth angle.

Specifically, the actual calculating process of the single star-based orientation method is provided, which is described as follows.

(S1) Any two orthogonal side surfaces of a hexahedron on a star sensor are denoted as a first side surface and a second side surface, respectively. A component of a vector ν_(right) on an X_(n)-axis of a quasi-horizontal reference system is denoted as t₄. A component of the vector ν_(right) on a Z_(n)-axis of the quasi-horizontal reference system is denoted as t₆. The vector ν^(right) is obtained by rotating a normal vector of the second side surface ν _(right) around the Z_(n)-axis of the quasi-horizontal reference system by a preset azimuth angle θ_(dir) ^(right).

(S2) The t₄ and t₆ are calculated through the following equations:

$\left\{ \begin{matrix} {t_{4} = {- \left( {{\sin\alpha{\sin\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)}} + {\cos\alpha{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)}}} \right)}} \\ {t_{6} = \left( {{\cos{\alpha sin}\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)} - {\sin\alpha{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)}}} \right)} \end{matrix} \right.$

wherein α is an angle between an X_(s)-axis of a dual-axis level sensor reference system and an X-axis of a hexahedron reference system on the star sensor. θ_(x) ^(pitch) is a measurement reading of the dual-axis level sensor on the X_(s)-axis; θ_(y) ^(pitch) is a measurement reading of the dual-axis level sensor on a Y_(s)-axis of the dual-axis level sensor reference system. Δθ_(x) ^(pitch) is a bias of the dual-axis level sensor on the X_(s)-axis. Δθ_(y) ^(pitch) is a bias of the dual-axis level sensor on the Y_(s)-axis.

(S3) An observed vector of a single astronomical object in the hexahedron reference system is denoted as ν_(PRI). A reference vector of the single astronomical object in an inertial reference system is denoted as ν_(GND), where the reference vector of the single astronomical object is obtained from a star catalog, and the ν_(GND) and ν_(PRI) satisfy the following equation:

R _(z)(θ_(z))·R _(x)(θ_(x))·R _(y)(θ_(y))·ν_(GND)=ν_(PRI)

where R_(y)(θ_(y)) denotes that the reference vector ν_(GND) rotates around a Y-axis of the hexahedron reference system by an azimuth angle θ_(y). R_(y)(θ_(x)) denotes that the reference vector ν_(GND) rotates around the X-axis of the hexahedron reference system by a pitch angle θ_(x). R_(z)(θ_(z)) denotes that the reference vector ν_(GND) rotates around a Z-axis of the hexahedron reference system by a roll angle θ_(z).

(S4) The pitch angle θ_(x) and the roll angle θ_(z) are calculated according to the t₄ and t₆ through the following equations:

$\left\{ {\begin{matrix} {\theta_{x} = {{- \arcsin}t_{6}}} \\ {\theta_{z} = {{- \arcsin}\left( {t_{4}/\cos\theta_{x}} \right)}} \end{matrix}.} \right.$

It should be noted that a transformation matrix R_(GND2PRI) from the quasi-horizontal reference system to the hexahedron reference system is described according to a rotation sequence of rotating around the Y-axis by the θ_(y), rotating around the X-axis by the θ_(x) and rotating around the Z-axis by the θ_(z) (2-1-3 rotation) to obtain relations as:

$\left\{ {\begin{matrix} {t_{4} = {\cos\theta_{x}\sin\theta_{z}}} \\ {t_{6} = {\sin\theta_{x}}} \end{matrix}.} \right.$

The equations for computing the pitch angle θ_(x) and the roll angle θ_(z) are obtained through an arcsin function in accordance with the above relations.

(S5) ν_(GND0)=R_(x)(−θ_(x))·R_(z) (−θ_(z))·ν_(PRI) is made. According to a rotation sequence of rotating around the Z-axis by the θ_(z), rotating around the X-axis by the θ_(x) and rotating around the Y-axis by the θ_(z) (3-1-2 rotation), ν_(GND) R_(y) (−θ_(y))·ν_(GND0) is obtained. The azimuth angle θ_(y) is calculated through the following equation to complete a single star-based orientation:

$\theta_{y} = {\arctan 2\left( {\frac{{v{1 \cdot v}6} - {v{3 \cdot v}4}}{{v4^{2}} + {v6^{2}}},\frac{{v{1 \cdot v}4} - {v{3 \cdot v}6}}{{v4^{2}} + {v6^{2}}}} \right)}$

where [ν1 ν2 ν3]^(T)=ν_(GND)·ν₁-ν3 are components of the ν_(GND) on the X-axis, the Y-axis and the Z-axis, respectively. [ν4 ν5 ν6]^(T)=ν_(GND0). ν4-ν6 are components of the ν_(GND0) on the X_(n)-axis, the Y_(n)-axis and the Z_(n)-axis, respectively.

A first calibrating process shown in FIG. 1 is a calibration of a transformational relationship between the quasi-horizontal reference system and the hexahedron reference system.

The quasi-horizontal reference system is constructed according to an angle information of a normal vector of the first side surface ν_(front) and the normal vector of the second side surface (considering that the first side surface is a front side surface and the second side surface is a right side surface). Referring to FIG. 2, the X_(n)-axis of the quasi-horizontal reference system is defined as a projection of the X-axis of the hexahedron reference system on a horizontal plane. A Y_(n)-axis of the quasi-horizontal reference system is defined such that the Y_(n)-axis of the quasi-horizontal reference system is constrained by orthogonality and lies in the horizontal plane, namely, the Y_(n)-axis is orthogonal to the X_(n)-axis on the horizontal plane and a Z_(n)-axis of the quasi-horizontal reference system satisfies a right-hand rule, and has a general upward pointing direction.

A normal pitch angle θ_(pitch) ^(front) of the first side surface and a normal pitch angle θ_(pitch) ^(front) of the second side surface are measured by a first theodolite and a second theodolite, respectively. An axis of a measuring lens barrel of the first theodolite coincides with a normal of the first side surface of the hexahedron, and an axis of a measuring lens barrel of the second theodolite coincides with a normal of the second side surface of the hexahedron. In the quasi-horizontal reference system, the normal vector ν_(front) is expressed as

$v_{front} = {{{R_{y}\left( \theta_{pitch}^{front} \right)} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} {vf1} \\ 0 \\ {vf3} \end{bmatrix}}$

where R_(y)(θ_(pitch) ^(front)) denotes that a vector [0 0 1] rotates around a Y_(n)-axis by the normal pitch angle θ_(pitch) ^(front). νƒ1, 0 and νƒ3 are components of the second normal vector ν_(front) on the X_(n)-axis, the Y_(n)-axis and the Z_(n)-axis, respectively. The normal vector ν _(right) without rotating by the preset azimuth angle θ_(dir) ^(right) is expressed as:

${{\overset{\_}{v}}_{right} = {{{R_{y}\left( \theta_{pitch}^{right} \right)} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} {vr1} \\ 0 \\ {vr3} \end{bmatrix}}},$

where R_(y)(θ_(pitch) ^(right)) denotes that the vector [00 1] rotates around the Y_(n)-axis of the quasi-horizontal reference system by the normal pitch angle θ_(pitch) ^(right). νr1, 0 and νr3 are components of the normal vector ν _(right) on the X_(n)-axis, Y_(n)-axis and Z_(n)-axis, respectively.

The vector ν_(right) is obtained by rotating the vector ν _(right) by the preset azimuth and the vector ν_(right) is expressed as follows:

$v_{right} = {{{R_{z}\left( \theta_{dir}^{right} \right)} \cdot \begin{bmatrix} {vr1} \\ 0 \\ {vr3} \end{bmatrix}} = \begin{bmatrix} {vr{1 \cdot \cos}\theta_{dir}^{right}} \\ {{- v}r{1 \cdot \sin}\theta_{dir}^{right}} \\ {vr3} \end{bmatrix}}$

where R_(z)(θ_(dir) ^(right)) denotes that a vector [νr1 0 νr3] rotates around the Z_(n)-axis by the preset azimuth angle θ_(dir) ^(right).

According to an orthogonal relationship between the normal vector ν_(front) and the normal vector ν_(right) (ν_(front) ⊥ ν_(right)), the following equation is obtained:

$\theta_{dir}^{right} = {{{acos}\left( \frac{{- v}f{3 \cdot v}r3}{vf{1 \cdot v}r1} \right)}.}$

A third normal vector ν_(up) of a third surface is obtained according to a cross product, expressed as

ν_(up)=ν_(right)×ν_(front).

Therefore, the transformation matrix R_(GND2PRI) is expressed as:

R _(GND2PRI)=[ν_(front)ν_(right)ν_(up)].

A second calibrating process shown in FIG. 1 is a calibration of a transformational relationship between the dual-axis level sensor reference system and the quasi-horizontal reference system.

Referring to FIG. 2, an angle between the X_(s)-axis and the X-axis is assumed as α. Since the dual-axis level sensor reference system is coplanar with the hexahedron reference system, an angle between the Y_(s)-axis and the X-axis is α+π/2. A direction vector of the X_(s)-axis in the hexahedron reference system is expressed as [cos α 0−sin α] and a direction vector of the Y_(s)-axis in the hexahedron reference system is expressed as [−sin α 0−cos α].

A transpose matrix of the transformation matrix R_(GND2PRI) is obtained, expressed as follows:

$R_{{PRI}2{GND}} = {R_{{GND}2{PRI}}^{T} = \begin{bmatrix} t_{1} & t_{2} & t_{3} \\ t_{4} & t_{5} & t_{6} \\ t_{7} & t_{8} & t_{9} \end{bmatrix}}$

where T represents a transposition operation. t₁-t₉ are elements of the transformation matrix R_(GND2PRI).

A direction vector ν_(x), of the X_(s)-axis in the quasi-horizontal reference system is acquired, expressed as follows:

$v_{X_{s}} = {\begin{bmatrix} {{t_{1}\cos\alpha} - {t_{3}\sin\alpha}} \\ {{t_{4}\cos\alpha} - {t_{6}\sin\alpha}} \\ {{t_{7}\cos\alpha} - {t_{9}\sin\alpha}} \end{bmatrix}.}$

A direction vector ν_(y), of the Y_(s)-axis in the quasi-horizontal reference system is acquired, expressed as follows:

$v_{Y_{s}} = {\begin{bmatrix} {{{- t_{1}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{3}\mspace{14mu}\cos\mspace{14mu}\alpha}} \\ {{{- t_{4}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\cos\mspace{14mu}\alpha}} \\ {{{- t_{7}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{9}\mspace{14mu}\cos\mspace{14mu}\alpha}} \end{bmatrix}.}$

The measurement reading of the direction vector ν_(x), on the dual-axis level sensor is θ_(x) ^(pitch). The measurement reading of the direction vector ν_(y), on the dual-axis level sensor is θ_(y) ^(pitch). The bias of the dual-axis level sensor on the Xs-axis is Δθ_(x) ^(pitch). The bias of the dual-axis level sensor on the Ys-axis is Δθ_(y) ^(pitch)

Relations between t₄ and t₆ according to the direction vector ν_(x), and the direction vector ν_(y), is built, expressed as follows:

$\left\{ {\begin{matrix} {{\sin\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)} = {{t_{4}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\sin\mspace{14mu}\alpha}}} \\ {{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)} = {{{- t_{4}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\cos\mspace{14mu}\alpha}}} \end{matrix}.} \right.$

According to the relations between t₄ and t₆, a relation between the measurement reading θ_(x) ^(pitch) and the bias Δθ_(x) ^(pitch) and a relation between the measurement reading θ_(y) ^(pitch) and the bias Δθ_(y) ^(pitch) are obtained, expressed as:

$\left\{ {\begin{matrix} {\theta_{x}^{pitch} = {{\Delta\theta_{x}^{pitch}} - {\arcsin\left( {{t_{4}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\sin\mspace{14mu}\alpha}} \right)}}} \\ {\theta_{y}^{pitch} = {{\Delta\theta_{y}^{pitch}} - {\arcsin\left( {{{- t_{4}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\cos\mspace{14mu}\alpha}} \right)}}} \end{matrix}.} \right.$

The star sensor is placed under a calibration device, as shown in FIG. 3. A two-axis level degree of the dual-axis level sensor is changed to obtain (θ_(x) ^(pitch), θ_(y) ^(pitch)) and (t₄, t₆) under different two-axis level degrees and then multiple mapping relations of (θ_(x) ^(pitch), θ_(x) ^(pitch))→(t₄, t₆) are obtained. The (θ_(x) ^(pitch), θ_(y) ^(pitch)) and (t₄, t₆) under different two-axis level degrees into the relation between the measurement reading θ_(x) ^(pitch) and the bias Δθ_(x) ^(pitch) and the relation between the measurement reading θ_(y) ^(pitch) and the bias Δθ_(y) ^(pitch) are plugged to obtain the angle α, the bias Δθ_(x) ^(pitch) and the bias Δθ_(y) ^(pitch).

In the orientation method provided herein, the hexahedron reference system is measured by means of the autocollimation of the theodolite, and the quasi-horizontal reference system is measured by the dual-axis level sensor. Then structural parameters of the quasi-horizontal reference system and the hexahedron reference system can be calibrated according to the pitch angle measured by the theodolite and the reading of the dual-axis level sensor. Accordingly, with the help of the theodolite, the collection of all calibrated data can be completed, simplifying the calibration process.

In summary, the single star-based orientation method provided herein can be described as follows.

The first calibrating process is performed as follows. Normal vectors of two orthogonal side surfaces on the hexahedron of the star sensor are measured to obtain the transformational relationship between the quasi-horizontal reference system and the hexahedron reference system. The X_(n)-axis of the quasi-horizontal reference system is defined as a projection of the X-axis of the hexahedron reference system in a horizontal plane. The Y_(n)-axis is orthogonal to the X_(n)-axis on the horizontal plane and the Z_(n)-axis of the quasi-horizontal reference system satisfies a vertically-upward right-hand rule.

The second calibrating process is performed as follows. By means of the mapping relation between the measurement reading of the dual-axis level sensor and normal measurements of the theodolites, the transformational relationship between the dual-axis level sensor reference system and the quasi-horizontal reference system is obtained. After the second calibrating process, the angle between an X_(s)-axis of a dual-axis level sensor reference system and an X-axis of a hexahedron reference system, the bias of the dual-axis level sensor on the X_(s)-axis, and the bias of the dual-axis level sensor on the Y_(s)-axis are obtained.

The actual calculating process is performed as follows. By means of an attitude information of the X_(s)-axis and the Y_(s)-axis obtained in the second calibrating process, the vectors expressed in the quasi-horizontal reference system is obtained through the observation vector of the single astronomical object. An azimuth angle difference between the quasi-horizontal reference system and the inertial reference system, that is orientation information, is obtained by combining with the reference vector of the single astronomical object in the inertial reference system from the star catalog.

This application eliminates an uncertainty of solution when solving for an attitude with a single vector by means of the information from the dual-axis level sensor. Furthermore, a quasi-horizontal reference system is constructed through the 2-1-3 rotation. Vectors of the single astronomical object are projected onto the quasi-horizontal reference system. By comparing the azimuth angle difference between a projected azimuth angle and an azimuth angle in the inertial reference system, the single star-based orientation of is completed.

Mentioned above are merely some embodiments of this disclosure, which are not intended to limit the disclosure. It should be understood that any changes, replacements and modifications made by those killed in the art without departing from the spirit and scope of this disclosure should fall within the scope of the present disclosure defined by the appended claims. 

What is claimed is:
 1. A single star-based orientation method using a dual-axis level sensor, comprising: (S1) denoting any two orthogonal side surfaces of a hexahedron on a star sensor as a first side surface and a second side surface, respectively; denoting a component of a vector ν_(right) on an X_(n)-axis of a quasi-horizontal reference system as t₄; and denoting a component of the vector ν_(right) on a Z_(n)-axis of the quasi-horizontal reference system as t₆; wherein the vector ν_(right) is obtained by rotating a normal vector of the second side surface around the Z_(n)-axis of the quasi-horizontal reference system by a preset azimuth angle θ_(dir) ^(right); (S2) calculating the t₄ and t₆ through the following equations: $\left\{ {\begin{matrix} {t_{4} = {- \left( {{\sin\mspace{14mu}\alpha\mspace{14mu}{\sin\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)}} + {\cos\mspace{14mu}\alpha\mspace{14mu}{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)}}} \right)}} \\ {t_{6} = \left( {{\cos\mspace{14mu}\alpha\mspace{14mu}{\sin\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)}} - {\sin\mspace{14mu}\alpha\mspace{14mu}{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)}}} \right)} \end{matrix};} \right.$ wherein α is an angle between an X_(s) axis of a dual-axis level sensor reference system and an X-axis of a hexahedron reference system on the star sensor; θ_(x) ^(pitch) is a measurement reading of the dual-axis level sensor on the X_(s)-axis; θ_(y) ^(pitch) is a measurement reading of the dual-axis level sensor on a Y_(s)-axis of the dual-axis level sensor reference system; Δθ_(x) ^(pitch) is a bias of the dual-axis level sensor on the X_(s)-axis; and Δθ_(y) ^(pitch) is a bias of the dual-axis level sensor on the Y_(s)-axis; (S3) denoting an observation vector of a single astronomical object in the hexahedron reference system as ν_(PRI); and denoting a reference vector of the single astronomical object in an inertial reference system as ν_(GND), wherein the reference vector of the single astronomical object is obtained from a star catalog, and the ν_(GND)and ν_(PRI) satisfy the following equation: R _(z)(θ_(z))·R _(x)(θ_(x))·R _(y)(θ_(y))·ν_(GND)=ν_(PRI); wherein R_(y)(θ_(y)) denotes that the reference vector ν_(GND) rotates around a Y-axis of the hexahedron reference system by an azimuth angle θ_(y); R_(x)(θ_(x)) denotes that the reference vector ν_(GND) rotates around the X-axis of the hexahedron reference system by a pitch angle θ_(x); and R_(z)(θ_(z)) denotes that the reference vector ν_(GND) rotates around a Z-axis of the hexahedron reference system by a roll angle θ_(z); (S4) calculating the pitch angle θ_(x) and the roll angle θ_(z) according to the t₄ and t₆ through the following equations: $\left\{ {\begin{matrix} {\theta_{x} = {{- \arcsin}\mspace{14mu} t_{6}}} \\ {\theta_{z} = {- {\arcsin\left( {{t_{4}/\cos}\mspace{14mu}\theta_{x}} \right)}}} \end{matrix};} \right.$ (S5) letting ν_(GND0)=R_(x)(−θ_(z))·R_(z)(−θ_(z))·ν_(PRI) and obtaining ν_(GND)=R_(y)(−θ_(y))·ν_(GND0); and calculating the azimuth angle θ_(y) through the following equation to complete a single star-based orientation: ${\theta_{y} = {\arctan\mspace{14mu} 2\left( {\frac{{v\;{1 \cdot v}\; 6} - {v\;{3 \cdot v}\; 4}}{{v4^{2}} + {v6^{2}}},\frac{{v\;{1 \cdot v}\; 4} + {v\;{3 \cdot v}\; 6}}{{v4^{2}} + {v6^{2}}}} \right)}};$ wherein [ν1 ν2 ν3]^(T)=ν_(GND); ν₁-ν3 are components of the ν_(GND) on three axes of the inertial reference system, respectively; [ν4 ν5 ν6]^(T)=ν_(GND0); and ν4-ν6 are components of the ν_(GND0) on the X_(n)-axis, the Y_(n)-axis and the Z_(n)-axis, respectively.
 2. The single star-based orientation method of claim 1, wherein in the step (S1), the preset azimuth angle θ_(dir) ^(right) is set through steps of: defining the X_(n)-axis of the quasi-horizontal reference system as a projection of the X-axis of the hexahedron reference system on a horizontal plane; defining a Y_(n)-axis of the quasi-horizontal reference system such that the Y_(n)-axis is orthogonal to the X_(n)-axis on the horizontal plane and a Z_(n)-axis of the quasi-horizontal reference system satisfies a right-hand rule, and has a general upward pointing direction; measuring, by a first theodolite and a second theodolite, a normal pitch angle θ_(pitch) ^(front) of the first side surface and a normal pitch angle θ_(pitch) ^(right) of the second side surface, respectively; wherein in the quasi-horizontal reference system, a normal vector ν_(front) of the first side surface is expressed as: ${v_{front} = {{{R_{y}\left( \theta_{pitch}^{front} \right)} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} {v\; f\; 1} \\ 0 \\ {v\;{f3}} \end{bmatrix}}};$ wherein R_(y)(θ_(pitch) ^(front)) denotes that a vector [0 0 1] rotates around a Y_(n)-axis of the quasi-horizontal reference system by the normal pitch angle θ_(pitch) ^(front); and νƒ1, 0 and νƒ3 are components of the normal vector ν_(front) on the X_(n)-axis, Y_(n)-axis and Z_(n)-axis, respectively; a normal vector ν _(right) of the second side surface without rotating by the preset azimuth angle θ_(dir) ^(right) is expressed as: ${{\overset{\_}{v}}_{right} = {{{R_{y}\left( \theta_{pitch}^{right} \right)} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} {v\; r\; 1} \\ 0 \\ {vr3} \end{bmatrix}}};$ wherein R_(y)(θ_(pitch) ^(right)) denotes that the vector [0 0 1] rotates around the Y_(n)-axis of the quasi-horizontal reference system by the normal pitch angle θ_(pitch) ^(right); and νr1, 0 and νr3 are components of the normal vector ν _(right) on the X_(n)-axis, Y_(n)-axis and Z_(n)-axis, respectively; supposing that the vector ν_(right) is obtained by rotating the vector ν _(right) by the preset azimuth angle θ_(dir) ^(right) and expressing the vector ν_(right) as follows: ${v_{right} = {{{R_{z}\left( \theta_{dir}^{right} \right)} \cdot \begin{bmatrix} {v\; r\; 1} \\ 0 \\ {vr3} \end{bmatrix}} = \begin{bmatrix} {{vr}\;{1 \cdot \cos}\mspace{14mu}\theta_{dir}^{right}} \\ {{- {vr}}\;{1 \cdot \sin}\mspace{14mu}\theta_{dir}^{right}} \\ {vr3} \end{bmatrix}}};$ wherein R_(z)(θ_(dir) ^(right)) denotes that a vector [νr1 0 νr3] rotates around the Z_(n)-axis by the preset azimuth angle θ_(dir) ^(right); and according to an orthogonal relationship between the normal vector ν_(front) and the normal vector ν_(right), obtaining the following equation: $\theta_{dir}^{right} = {a\mspace{14mu}{{\cos\left( \frac{{- {vf}}\;{3 \cdot {vr}}\; 3}{{vf}\;{1 \cdot {vr}}\; 1} \right)}.}}$
 3. The single star-based orientation method of claim 2, wherein an axis of a measuring lens barrel of the first theodolite coincides with a normal of the first side surface of the hexahedron, and an axis of a measuring lens barrel of the second theodolite coincides with a normal of the second side surface of the hexahedron.
 4. The single star-based orientation method for orientation of claim 1, wherein in the step (S2), the angle α, the bias Δθ_(x) ^(pitch) and the bias Δθ_(y) ^(pitch) are obtained through steps of: defining the X_(n)-axis of the quasi-horizontal reference system as a projection of the X-axis of the hexahedron reference system on a horizontal plane; defining a Y_(n)-axis of the quasi-horizontal reference system such that the Y_(n)-axis is orthogonal to the X_(n)-axis on the horizontal plane and the Z_(n)-axis of the quasi-horizontal reference system satisfies a vertically-upward right-hand rule; allowing the dual-axis level sensor reference system to be coplanar with the hexahedron reference system; and expressing a direction vector of the X_(s)-axis in the hexahedron reference system as [cos α 0−sin α] and a direction vector of the Y_(s)-axis in the hexahedron reference system as [−sin α 0−cos α]; denoting a normal vector of the first side surface as ν_(front); and obtaining a normal vector ν_(up), of a third side surface orthogonal to both the first side surface and the second side surface by cross product, expressed as: ν_(up)=ν_(right)×ν_(front); obtaining a transformation matrix R_(GND2PRI) from the quasi-horizontal reference system to the hexahedron reference system, expressed as: R _(GND2PRI)=[ν_(front)ν_(right)ν_(up)]; obtaining a transpose matrix of the transformation matrix R_(GND2PRI), expressed as: ${R_{PRI2GND} = {{R^{T}}_{{GND}\; 2{PRI}} = \begin{bmatrix} t_{1} & t_{2} & t_{3} \\ t_{4} & t_{5} & t_{6} \\ t_{7} & t_{8} & t_{9} \end{bmatrix}}};$ wherein T represents a transposition operation; and t₁-t₉ are elements of the transformation matrix R_(GND2PRI); acquiring a direction vector ν_(x), of the X_(s)-axis in the quasi-horizontal reference system, expressed as: ${v_{X_{s}} = \begin{bmatrix} {{t_{1}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{3}\mspace{14mu}\sin\mspace{14mu}\alpha}} \\ {{t_{4}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\sin\mspace{14mu}\alpha}} \\ {{t_{7}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{9}\mspace{14mu}\sin\mspace{14mu}\alpha}} \end{bmatrix}};$ acquiring a direction vector ν_(y), of the Y_(s)-axis in the quasi-horizontal reference system, expressed as: ${v_{Y_{s}} = \begin{bmatrix} {{{- t_{1}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{3}\mspace{14mu}\cos\mspace{14mu}\alpha}} \\ {{{- t_{4}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\cos\mspace{14mu}\alpha}} \\ {{{- t_{7}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{9}\mspace{14mu}\cos\mspace{14mu}\alpha}} \end{bmatrix}};$ building relations between t₄ and t₆ according to the direction vector ν_(x), and the direction vector ν_(y), expressed as: $\left\{ {\begin{matrix} {{\sin\left( {{\Delta\theta_{x}^{pitch}} - \theta_{x}^{pitch}} \right)} = {{t_{4}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\sin\mspace{14mu}\alpha}}} \\ {{\sin\left( {{\Delta\theta_{y}^{pitch}} - \theta_{y}^{pitch}} \right)} = {{{- t_{4}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\cos\mspace{14mu}\alpha}}} \end{matrix};} \right.$ according to the relations between t₄ and t₆, obtaining a relation between the measurement reading θ_(x) ^(pitch) and the bias Δθ_(x) ^(pitch) and a relation between the measurement reading θ_(y) ^(pitch) and the bias Δθ_(y) ^(pitch), expressed as: $\left\{ {\begin{matrix} {\theta_{x}^{pitch} = {{\Delta\theta_{x}^{pitch}} - {\arcsin\left( {{t_{4}\mspace{14mu}\cos\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\sin\mspace{14mu}\alpha}} \right)}}} \\ {\theta_{y}^{pitch} = {{\Delta\theta_{y}^{pitch}} - {\arcsin\left( {{{- t_{4}}\mspace{14mu}\sin\mspace{14mu}\alpha} - {t_{6}\mspace{14mu}\cos\mspace{14mu}\alpha}} \right)}}} \end{matrix};} \right.$ changing a two-axis level degree of the dual-axis level sensor to obtain (θ_(x) ^(pitch), θ_(y) ^(pitch)) and (t₄, t₆) under different two-axis level degrees; and plugging the (θ_(x) ^(pitch), θ_(y) ^(pitch)) and (t₄, t₆) under different two-axis level degrees into the relation between the measurement reading θ_(x) ^(pitch) and the bias Δθ_(x) ^(pitch) and the relation between the measurement reading θ_(y) ^(pitch) and the bias Δθ_(y) ^(pitch) to obtain the angle α, the bias Δθ_(x) ^(pitch) and the bias Δθ_(y) ^(pitch). 